Study Guide: Inference for Means and Proportions (Chapters 16–20)

Inference for Means and Proportions

Overview

This study guide covers key concepts and procedures from Chapters 16–20, focusing on statistical inference for means and proportions, including confidence intervals and hypothesis testing. These topics are foundational for understanding how to draw conclusions about populations based on sample data.

Confidence Intervals for Population Means

Constructing Confidence Intervals

• Population Mean (μ): The average value in the entire population.

• Sample Mean (\(\bar{x}\)): The average value in the sample.

• Population Standard Deviation (σ): The spread of values in the population.

• Sample Standard Deviation (s): The spread of values in the sample.

• Sample Size (n): The number of observations in the sample.

When σ is unknown (which is common), use the t-distribution:

$\text{Confidence Interval} = \bar{x} \pm t^* \left( \frac{s}{\sqrt{n}} \right)$

• t* is the critical value from the t-distribution with \(n-1\) degrees of freedom.

• Use statistical software or tables to find t* for the desired confidence level (e.g., 95%).

Example: For a sample mean of 50, sample standard deviation of 10, and n = 25, the 95% confidence interval is:

$50 \pm t^* \left( \frac{10}{\sqrt{25}} \right) = 50 \pm t^* (2)$

Find t* for 24 degrees of freedom (from tables or calculator).

Confidence Intervals for Proportions

Constructing Confidence Intervals for a Proportion

• Sample Proportion (\(\hat{p}\)): The proportion of successes in the sample.

• Standard Error (SE): The estimated standard deviation of the sample proportion.

$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

$\text{Confidence Interval} = \hat{p} \pm z^* \cdot SE$

• z* is the critical value from the standard normal distribution (e.g., 1.96 for 95% confidence).

Example: If \(\hat{p} = 0.6\) and n = 100, then

$SE = \sqrt{\frac{0.6 \times 0.4}{100}} = 0.049$

$\text{Confidence Interval} = 0.6 \pm 1.96 \times 0.049 = (0.504, 0.696)$

Hypothesis Testing for Means

Steps in Hypothesis Testing

1. State the Hypotheses:

   • Null hypothesis (H0): μ = μ0

   • Alternative hypothesis (Ha): μ ≠ μ0, μ > μ0, or μ < μ0

2. Calculate the Test Statistic:

   $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

3. Find the p-value: Use statistical software or t-tables.

4. Make a Decision: Compare the p-value to α (significance level, e.g., 0.05).

5. State the Conclusion: Reject or fail to reject H0.

Example: Testing if the mean is greater than 100 with sample mean 105, s = 15, n = 36:

$t = \frac{105 - 100}{15/\sqrt{36}} = \frac{5}{2.5} = 2.0$

Find the p-value for t = 2.0 with 35 degrees of freedom.

Hypothesis Testing for Proportions

Steps in Hypothesis Testing

1. State the Hypotheses:

   • Null hypothesis (H0): p = p0

   • Alternative hypothesis (Ha): p ≠ p0, p > p0, or p < p0

2. Calculate the Test Statistic:

   $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

3. Find the p-value: Use statistical software or normal tables.

4. Make a Decision: Compare the p-value to α.

5. State the Conclusion: Reject or fail to reject H0.

Example: Testing if the proportion is less than 0.5 with \(\hat{p} = 0.45\), n = 200:

$z = \frac{0.45 - 0.5}{\sqrt{\frac{0.5 \times 0.5}{200}}} = \frac{-0.05}{0.035} = -1.43$

Find the p-value for z = -1.43.

Comparing Two Means or Proportions

Confidence Interval for the Difference of Two Means

$\text{CI} = (\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

• Use when comparing means from two independent samples.

Confidence Interval for the Difference of Two Proportions

$\text{CI} = (\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Use when comparing proportions from two independent samples.

Hypothesis Test for the Difference of Two Means

$t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Null hypothesis: μ1 = μ2

• Alternative hypothesis: μ1 ≠ μ2, μ1 > μ2, or μ1 < μ2

Hypothesis Test for the Difference of Two Proportions

$z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

• Where \(\hat{p}\) is the pooled proportion: $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$

Interpreting Results

• Reject H0: There is evidence for the alternative hypothesis.

• Fail to Reject H0: There is not enough evidence for the alternative hypothesis.

• Always state your conclusion in the context of the problem.

Using Statistical Software (e.g., Calculator or StatCrunch)

• Use STAT, CALC, NORMAL or T-intervals for confidence intervals.

• Use STAT, TESTS for hypothesis testing.

• Input sample statistics as required (mean, SD, n, proportions, etc.).

• Interpret output: confidence interval bounds, test statistic, p-value, and conclusion.

Summary Table: Key Procedures

Procedure	Formula	When to Use
CI for Mean	$\bar{x} \pm t^* \left( \frac{s}{\sqrt{n}} \right)$	Unknown σ, sample mean and SD
CI for Proportion	$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	Sample proportion
Test for Mean	$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$	Compare sample mean to hypothesized mean
Test for Proportion	$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$	Compare sample proportion to hypothesized proportion
CI for Difference of Means	$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$	Compare two independent means
CI for Difference of Proportions	$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$	Compare two independent proportions

Additional Info

• Degrees of freedom for t-tests are typically n-1 for one sample, or calculated using a formula for two samples.

• Always check assumptions: random sampling, independence, and normality (for means).

• For proportions, check that np and n(1-p) are both at least 5.